Commutative rings with identity have ring topologies

Directed Graphs of Commutative Rings with Identity

The directed graph of a ring is a graphical representation of its additive and multiplicative structure. Using the directed edge relationship (a, b) → (a + b, a · b), one can create a directed graph for every ring. This paper focuses on the structure of the sources in directed graphs of commutative rings with identity, with special concentration in the finite and reduced cases. Acknowledgements...

Commutative Γ-rings Do Not Model All Commutative Ring Spectra

We show that the free E∞-algebra on a zero-cell cannot be modeled by a commutative Γ-ring. The proof shows that DyerLashof operations of positive degree must vanish on the zero’th homology of such an object.

Tiling with Commutative Rings

so that every square is covered by exactly one domino? In other words, can R be tiled by vertical and horizontal dominoes? The coloring gives the answer to this well-known problem away. The region R has 32 black squares and 30 white squares. Since each domino covers exactly one black and one white square, no tiling is possible. The aim of this article is to explain a way to tackle tiling proble...

-homology of Commutative Rings and of E1 Ring Spectra

ON COMMUTATIVE GELFAND RINGS

A ring is called a Gelfand ring (pm ring ) if each prime ideal is contained in a unique maximal ideal. For a Gelfand ring R with Jacobson radical zero, we show that the following are equivalent: (1) R is Artinian; (2) R is Noetherian; (3) R has a finite Goldie dimension; (4) Every maximal ideal is generated by an idempotent; (5) Max (R) is finite. We also give the following resu1ts:an ideal...